# Questions tagged [dual-spaces]

The dual space of a vector space $V$ over a field $k$ is the vector space of all linear maps from $V$ into $k$.

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### If all compositions of a vector-valued function with its linear functionals are analytic, is the function itself analytic?

This is taken from Conway's A course in functional Analysis (p. 198, Exercise 4): Let $\mathscr{X}$ be a Banach space and $G \subset \mathbb{C}$ open. If $f: G \rightarrow \mathscr{X}$ is such that ...
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### Proof: If T (T') is surjective then T' (T) is injective.

I am a little unsure about my proof. I wanted to ask if this proof is correct.I would appreciate any suggestions and improvements. Futhermore I dont yet fully understand why in (a) from the ...
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### Is there any bounded linear map $T:X\to X$ such that $T$ is injective, $T^{\ast}$ is not surjective and $R(T^{\ast})$ is closed?

Here $X$ is Banach space and $T^{\ast}$ is the continuous dual(adjoint) of $T,R(T^{\ast})$ stands for the range of $T^{\ast}.$ The following is what I get: If such a $T$ exists, then $X$ cannot be ...
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### Are the weakly bounded subsets of the double dual bounded for the natural topology?

I'm trying to get a better understanding of the topologies on the double dual of a Hausdorff locally convex space (l.s.c.). The following question has then come up, and I'm unable to find an answer. ...
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I found the following exercise in my lecture notes: Let $\mathcal{A}$ be a $C^*$-algebra. Then every multiplicative functional is an extremal point of $K_1^{\mathcal{A}'}(0)$. Other than writing $m = ... 1 vote 0 answers 33 views ### Weak*-separability of the unit ball of$X’$and density characters and cardinalities of$X$and$X’$Let$X$be a Banach space,$X’$be its continuous dual such that its unit ball is weak*-separable. I’ve been wondering what can be said about the density characters (which I’ll denote by$d$) and ... 0 votes 0 answers 32 views ### How can a vector dual space be used to model division? A couple of months ago I posted a question about how to formally define systems of units, and someone posted this fascinating response, explaining that each base unit can be the basis of a single ... 0 votes 1 answer 63 views ### If$L_n \rightarrow L \in X^*$in the weak* sense and$x_n \rightarrow x$in norm, does$L_n(x_n) \rightarrow L(x)$? Let$X$be a Banach space and$X^*$its dual. Let$\{L_n\}$be a sequence in$X^*$such that$L_n \rightarrow L$in the weak* sense, and suppose$\{x_n\}$is a sequence in$X$such that$x_n \...
Let $X$ be a locally convex space, whose topology is defined by a family of seminorms $\mathcal{P}$. Fact. For every continuous seminorm $q: X\to \mathbb{F}$ and positive number $\epsilon>0$, the ...